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In this post we will give a heuristic derivation of a result of Vershik, describing the shape of a random partition of a large integer . (Vershik’s Russian original is available here; English translation is pay-walled.)

By a partition of , we mean positive integers with . We draw a partition as a collection of boxes: For example, this is :

Suppose we let , select partitions of uniformly at random and rescale the size of the boxes by , so that the diagram of the partition always has area . What is the shape of the most likely diagram?

Update — there are now not one, but two, positions available! The application has been extended to the end of November.

We’ve just put up an ad for a new 2 year postdoctoral position at the ANU, to work with myself and Tony Licata. We’re looking for someone who’s interested in operator algebras, quantum topology, and/or representation theory, to collaborate with us on Australian Research Council funded projects.

The ad hasn’t yet been crossposted to MathJobs, but hopefully it will eventually appear there! In any case, applications need to be made through the ANU website. You need to submit a CV, 3 references, and a document addressing the selection criteria. Let me know if you have any questions about the application process, the job, or Canberra!

Those of you who are interested in college math instruction may be interested in a no-longer-so-new blog “Michigan Math In Action”, which a number of our faculty started last year. (I was involved in the sense of telling people “blogs are fun!”, but haven’t written anything for them yet.) It mostly features thoughtful pieces on teaching calculus and similar courses.

Recently, Gavin Larose put up a lengthy footnoted post on the effort that goes into running our “Gateway testing” center, and the benefits we get from it. This is a room designed for proctoring computerized tests of basic skills, and we use it for things like routine differentiation or putting matrices into reduced row echelon form, which we want every student to know but which are a waste of class time. Check it out!

Just a quick reminder that, if you are looking for graduate support to attend ALGECOM at the University of Michigan on Saturday October 24, or to register for the poster session, you should please send an e-mail to speyer@umich.edu by Tuesday Sept 15. (Yes, after sunset but before midnight is fine, I won’t be online during Rosh Hoshanah either.)
Even if you are not requesting support, I’d appreciate knowing that you are coming.

Let be a finite group, and let be a positive integer dividing . Then the number of solutions to in is divisible by .

This is a 1907 theorem of Frobenius. Along with the Sylow theorems, it is one of the few nontrivial elementary results about a completely general finite group. And it has some nice applications, which you can read about on Mathoverflow. But it has never made it into the standard basic group theory syllabus the way the Sylow theorems have. I wanted to give it as a challenging problem last time I taught group theory, but I didn’t find a proof that I liked enough.

The last few days, I’ve been thinking about the problem again, and I found what I think is a decent counting proof. I have the feeling there is a really slick proof in here waiting to get out. Let me know if you can find it!

There aren’t many blog posts about vertex operator algebras, so I thought I’d help fill this gap by mentioning a substantial advance by Jethro van Ekeren, Sven Möller, and Nils Scheithauer that appeared on the ArXiv last month. The most important feature is that this paper resolves several folklore conjectures that have been around since near the beginning of vertex operator algebra theory. This was good for me, since I was able to use some of these results to prove the Generalized Moonshine Conjecture much more quickly than I had expected. I won’t say much about moonshine here, as I think it deserves its own post.

The University of Michigan at Ann Arbor is proud to be hosting
ALGECOM, the twice annual midwestern conference on algebra, geometry
and combinatorics on Saturday, October 24. We will feature four
speakers, namely,

as well as a poster session. If you would like to submit a poster, please e-mail (David Speyer) with a quick summary of your work by September 15.

A block of rooms has been reserved at the (Lamp Post Inn) under the name of ALGECOM.

This conference is supported by a conference grant form the NSF. Limited funds are available for graduate student travel to the conference. Please contact (David Speyer) to request support, and include a note from your adviser.

More information will be added to our website as it becomes available.

A number of blogs I read are arguing about a paradox, posed by tumblr blogger perversesheaf. Here is my attempt to explain what the paradox says.

Suppose that a drug company wishes to create evidence that a drug is beneficial, when in fact its effect is completely random. To be concrete, we’ll say that the drug has either positive or negative effect for each patient, each with probability . The drug company commits in advance that they will state exactly what their procedure will be, including their procedure for when to stop tasks, and that they will release all of their data. Nonetheless, they can guarantee that a Bayesian analyst with a somewhat reasonable prior will come to hold a strong belief that the drug does some good. Below the fold, I’ll explain how they do this, and think about whether I care.

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Secret Blogging Seminar

A group blog by 8 recent Berkeley mathematics Ph.D.'s. Commentary on our own research, other mathematics pursuits, and whatever else we feel like writing about on any given day. Sort of like a seminar, but with (even) more rude commentary from the audience.